Extension of the Fermi–Amaldi functional to fractional electron number
Bibliographic record
Abstract
The Fermi-Amaldi correction is a conceptually significant density-functional approximation with explicit dependence on the electron number N. This dependence raises the question of whether N should be treated as the integral of the total electron density for systems with noninteger (fractional) N. We answer this question negatively by observing that the Fermi-Amaldi functional is identical with the exact exchange functional for any number of fermions or bosons occupying the same spatial orbital and in systems where occupied orbitals have zero differential overlap. This conclusion is consistent with previous reports that the preservation of certain properties of N-dependent functionals requires treating N as a parameter. Our analysis also suggests that the Fermi-Amaldi functional is discontinuous within a given spin-channel of the total electron density and continuous on the boundaries between different spin-channels at N = 1. This implies that the jump discontinuity of the exact exchange-correlation potential observed at N = 1 is a correlation effect. The last conclusion is illustrated with numerical calculations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".